Uniqueness theorem for power series

[snipez90]

Hi, for awhile I was agonizing over part b) of this http://books.google.com/books?id=WZ...complex analysis&pg=PA62#v=onepage&q&f=false" of Theorem 3.2 in Lang's Complex Analysis.

But I think part of the reason was that I kept concentrating on the second sentence of the theorem statement in part a), instead of the entire statement. Just to make sure, part b) follows by using the contrapositive of part a) so that h reduces to a constant which is in fact 0, correct? Thanks.

 

[Landau]

Yep. Since h(x)=0 for all x in an infinite set with 0 as accumulation point, the second sentence of part (a) applied to h is NOT true. Hence (by contraposition) the first sentence of part (a) is not true, meaning h is constant. Finally, h being constant and h(x)=0 for some x, it follows that h is everywhere 0.